Paints typically comprise pigments dispersed within a binder. The pigments may comprise i) weakly scattering or substantially non-scattering coloured pigments that substantially do not scatter light and selectively absorb different wavelengths of visible light to impart colour ii) substantially non-absorbing pigments that scatter light and iii) strongly scattering coloured pigments that scatter light and selectively absorb different wavelengths of visible light to impart colour.
The Kubelka Munk equation is a radiation transfer equation commonly used in the field of paints and pigments to characterise the visible reflection spectrum of materials. The use of the Kubelka Munk method when describing multi-component pigment mixtures is usually based on equations of the form:
      (          K      S        )    =                              k          1                ×                  φ          1                    +                        k          2                ×                  φ          2                    +                        k          3                ×                  φ          3                ⁢                                  ⁢        …                                      s          1                ×                  φ          1                    +                        s          2                ×                  φ          2                    +                        s          3                ×                  φ          3                ⁢                                  ⁢        …            
where (K/S) is the ratio of the Kubelka Munk absorption and scattering coefficients and ki, si and φi are the absorption coefficient, scattering coefficient and volume fraction for an individual pigment. The above equation is inaccurate and problematic at high volume fractions of scattering pigments due to crowding effects (for example, for conventional titanium dioxide pigments, in the visible part of the spectrum, inaccuracies are apparent at pigment volume concentrations greater than about 15%). Crowding effects reduce the scattering efficiency of a scattering pigment due to a high density of pigment particles in a particular volume, thus the relationship between scattering efficiency and pigment volume concentration becomes non linear.
The particle volume concentration at which crowding effects occur is different for different pigments and different wavelengths. As the particle size of the pigment increases or wavelength decreases, the volume fraction above which the crowding effect takes effect is increased. Thus, when considering the near infrared part of the spectrum, the pigment volume concentration below which the relationship between scattering efficiency and pigment volume concentration is linear, occurs at even lower pigment volume concentrations than when considering the visible part of the spectrum.
One traditional approach to characterising the absorption and scattering coefficients of a pigment in the visible part of the spectrum is to measure reflectivity when incorporated into carbon black and titanium dioxide pigment pastes. The extension of this approach into the near infrared part of the spectrum is problematic, especially for pigments where both scattering and absorption are significant. The opacity of a conventional titanium dioxide paste falls rapidly as the wavelengths of interest are increased into the near infrared.
To obtain an optically thick system in the near infrared using conventional paint film thicknesses (<300 nm) thus requires volume fractions of pigment above which the crowding effect needs to be taken account of. At such volume fractions the addition of a significant volume fraction of a second scattering particle will lead to a more complex relationship between the Kubelka Munk scattering coefficient S and the volume fraction of the second scattering particle φ2 than that when crowding effects can be ignored.S≠φ1s1+φ2ss S=S(φ1,φ2)